algorithms – Huffman Example and one solved-question and playing with words

I read an solved-old exam True-False Question and mark the following as False:

for a text that it’s characters get from set of $n$ alphabets then average length of codes is $O(log n)$.

My question is why this is False? (please attention that not says
average of codewords).

I guess Fibonacci series and Huffman tree, but really I confused on basic concept again.

for example I read that “the average codeword length in a Huffman tree
is $Omega (log n)$. really confusing and playing with word is very
bad “codeword” vs “code” vs… and… anyone can help me and clear me in a
very basic and short answer about my presented facts?